1. Field of the Invention
The present invention relates to adaptive control methods and apparatus for automatically optimizing a control gain. More particularly, the invention relates to adaptive control methods and apparatus preferred in controlling cross-machine direction profiles in a paper machine.
The effect of feedback control varies greatly, depending on a combination of the period of a disturbance present in a process and a control gain. In other words, there are two types of disturbance, one of which has such a period as to enable the disturbance to be attenuated by feedback control and the other has such a period as to cause the disturbance to be amplified instead as the result of feedback control.
Conventionally, the tuning of control gains has been achieved by trial and error in the field, while observing controllability against cyclic disturbances present in a process. This conventional tuning method has been problematic, however. If a control gain is decreased in an attempt to prevent the amplitude of a short-period disturbance from increasing as the result of control, controllability against long-period disturbances or follow-up capability for setpoint changes will deteriorate. It is therefore desirable that the control gain be increased as much as possible within the given tolerance limits. On the other hand, it is difficult to search for the magnitude of control gain that gives the optimum balance.
Another problem is that the period and/or amplitude of disturbance present in a process varies with the state thereof. The interval of such variation is wide-ranging, from several hours to a few days, or even to a few months. This means that the control gain that is optimum at some point in time may be improper several hours or several days later. Consequently, controllability would deteriorate and in some cases the control gain must be re-tuned.
A general object of the present invention, therefore, is to provide adaptive control methods and apparatus whereby a control gain can be automatically optimized online.
2. Description of the Prior Art
FIG. 1 is a diagrammatic view showing the configuration of a paper machine. In the figure, produced paper 61 is smoothed in its entirety and tuned in its thickness profile by a calender 62, and wound onto a reel 64. A sensor 63 is placed immediately before the reel 64 to measure the moisture percentage and/or thickness of the produced paper 61. A measurement signal detected by the sensor 63 is then input to a measurement computation unit 65, where the profile of the signal is calculated. This profile is then input to the control unit 66. The control unit 66 controls the paper machine according to the profile.
The sensor 63 scans the paper 61 in the cross-machine direction (from this side to the far side of the paper in the figure) to measure the moisture percentage and/or thickness. Since there are as many as 1200 measurement points in the cross-machine direction, the width of the paper is divided into multiple zones and the measured value of the midpoint of each zone is defined as the representative data point of that zone.
FIG. 2 is a schematic view showing the relationship between zones and measurement points. In FIG. 2, a numeral 71 indicates the way paper 61 is cut into rectangular slices and thus divided into N zones. A numeral 72 indicates the correspondence of these zones with measurement points. A plurality of measurement points are included in each zone (zone i), and the midpoint PC(i) among the plurality of measurement points is defined as the representative data point of that zone.
FIG. 3 is a block diagram showing the configuration of a system for controlling cross-machine direction profiles in a paper machine. In the figure, a setpoint variable R(s) and a control output variable C(s) (e.g., the measured value of paper thickness) of profile control are input to a calculation unit 81, where a deviation variable E(s) which is a difference between the setpoint and control output variables is calculated. The deviation variable E(s) is input to a controller 82 of finite settling-time response control type for calculation and output of a manipulated variable W(s). The manipulated variable W(s) is input through a hold unit 83 to a process 84 that can be approximated using a dead time and a first-order delay. Consequently, the process 84 is placed under feedback control.
A symbol V(s) denotes cyclic disturbance present in the process 84. Such cyclic disturbance includes interference due to a concentration or liquid level change in a system for blending various types of raw material or interference arising in the form of cyclic variations in the measurement signal of moisture percentage or paper thickness caused by an eccentricity in a rapidly rotating wire or roll. These cyclic disturbances have two types of period, one of which enables the disturbance to be attenuated by feedback control and the other causes the disturbance to be amplified instead as the result of feedback control.
Now the method of controlling a paper machine is described with reference to FIG. 3. The transfer functions of the hold unit 83 and process 84 that can be approximated using a dead time and a first-order delay are represented by equations (1) and (2) below:
Transfer function of hold unit 83                               H          ⁡                      (            s            )                          =                              1            -                          ⅇ                              -                Ts                                              s                                    (        1        )            
Transfer function of process 84                               P          ⁡                      (            s            )                          =                              Ke                          -              Ls                                            1            +                                          T                0                            ⁢              s                                                          (        2        )            
where
K=Process gain
T=Sampling interval
T0=Time constant
L=Dead time (L=mT, where m is 0 or a natural number).
Assuming HP(s)=H(s)P(s), then
xe2x80x83HP(s)G(s)(xe2x88x92C(s))+V(s)=C(s)
holds true from FIG. 3. Changing this equation gives
V(s)=(1+HP(s)G(s))xc2x7C(s)
From this equation, the control output C(s) is determined as                               C          ⁡                      (            s            )                          =                              1                          1              +                                                HP                  ⁡                                      (                    s                    )                                                  ⁢                                  G                  ⁡                                      (                    s                    )                                                                                ⁢                      V            ⁡                          (              s              )                                                          (        3        )            
Z-transforming equation (3) results in equation (4) below.                               C          ⁡                      (            z            )                          =                              1                          1              +                                                HP                  ⁡                                      (                    z                    )                                                  ⁢                                  G                  ⁡                                      (                    z                    )                                                                                ⁢                      V            ⁡                          (              z              )                                                          (        4        )            
From equations (1) and (2) noted above, z-transforming HP(s) gives equation (5) below.                                                                         HP                ⁡                                  (                  z                  )                                            =                                                [                                                                                    1                        -                                                  ⅇ                                                      -                            Ts                                                                                              s                                        ·                                                                  Ke                                                  -                          Ls                                                                                            1                        +                                                                              T                            0                                                    ⁢                          s                                                                                                      ]                                =                                                      K                    ⁡                                          (                                              1                        -                                                  z                                                      -                            1                                                                                              )                                                        ⁢                                                            z                                              -                        m                                                              ·                                          Z                      ⁡                                              [                                                  1                                                      s                            ⁡                                                          (                                                              1                                +                                                                                                      T                                    0                                                                    ⁢                                  s                                                                                            )                                                                                                      ]                                                                                                                                                                    =                                                                    K                    ⁡                                          (                                              1                        -                                                  z                                                      -                            1                                                                                              )                                                        ⁢                                                            z                                              -                        m                                                              ·                                          Z                      ⁡                                              [                                                                              1                            s                                                    -                                                      1                                                          s                              +                                                              T                                0                                                                  -                                  1                                                                                                                                                                    ]                                                                                            =                                                      K                    ⁡                                          (                                              1                        -                                                  z                                                      -                            1                                                                                              )                                                        ⁢                                                            z                                              -                        m                                                              ⁡                                          (                                                                        1                                                      1                            -                                                          z                                                              -                                1                                                                                                                                    -                                                  1                                                      1                            -                                                          ⅇ                                                                                                                                    -                                    T                                                                    /                                                                      T                                    0                                                                                                  ⁢                                                                  z                                                                      -                                    1                                                                                                                                                                                                                          )                                                                                                                                              =                                                                    K                    ⁡                                          (                                              1                        -                        α                                            )                                                        ⁢                                      z                                          -                                              (                                                  m                          +                          1                                                )                                                                                                              1                  -                                      α                    ⁢                                          xe2x80x83                                        ⁢                                          z                                              -                        1                                                                                                                                                                                    where                ⁢                                  xe2x80x83                                ⁢                α                            =                                                ⅇ                                                            -                      T                                        /                                          T                      0                                                                      .                                                                        (        5        )            
The transfer function G(z) of a controller of finite settling-time response control type is given by equation (6) below:                               G          ⁡                      (            z            )                          =                              1                                          K                *                            ⁡                              (                                  1                  -                                      α                    k                                                  )                                              ⁢                      xe2x80x83                    ⁢                                                    (                                  1                  -                                                            α                      k                                        ⁢                                          z                                              -                        k                                                                                            )                            ⁢                              (                                  1                  -                                      α                    ⁢                                          xe2x80x83                                        ⁢                                          z                                              -                        1                                                                                            )                                                                    (                                  1                  -                                      α                    ⁢                                          xe2x80x83                                        ⁢                                          z                                              -                        1                                                                                            )                            -                                                                    (                                          1                      -                      α                                        )                                                        (                                          1                      -                                              α                        k                                                              )                                                  ⁢                                                      z                                          -                                              (                                                  m                          +                          1                                                )                                                                              ⁡                                      (                                          1                      -                                                                        α                          k                                                ⁢                                                  z                                                      -                            k                                                                                                                )                                                                                                          (        6        )            
where K* denotes a control gain.
Now let us define a symbol g as g=K/K* (g greater than 0) and refer to g as a control gain ratio. From equations (5) and (6), we obtain equation (7) below:                                                                         1                +                                                      HP                    ⁡                                          (                      z                      )                                                        ⁢                                      G                    ⁡                                          (                      z                      )                                                                                  =                              1                +                                                                                                    K                        ⁡                                                  (                                                      1                            -                            α                                                    )                                                                    ⁢                                              z                                                  -                                                      (                                                          m                              +                              1                                                        )                                                                                                                                      1                      -                                              α                        ⁢                                                  xe2x80x83                                                ⁢                                                  z                                                      -                            1                                                                                                                                ⁢                                      1                                                                  K                        *                                            ⁡                                              (                                                  1                          -                                                      α                            k                                                                          )                                                                              ⁢                                      xe2x80x83                                    ⁢                                                                                    (                                                  1                          -                                                      α                            ⁢                                                          xe2x80x83                                                        ⁢                                                          z                                                              -                                1                                                                                                                                    )                                            ⁢                                              (                                                  1                          -                                                                                    α                              k                                                        ⁢                                                          xe2x80x83                                                        ⁢                                                          z                                                              -                                k                                                                                                                                    )                                                                                                            (                                                  1                          -                                                      α                            ⁢                                                          xe2x80x83                                                        ⁢                                                          z                                                              -                                1                                                                                                                                    )                                            -                                                                                                    (                                                          1                              -                              α                                                        )                                                                                (                                                          1                              -                                                              α                                k                                                                                      )                                                                          ⁢                                                                              z                                                          -                                                              (                                                                  m                                  +                                  1                                                                )                                                                                                              ⁡                                                      (                                                          1                              -                                                                                                α                                  k                                                                ⁢                                                                  z                                                                      -                                    k                                                                                                                                                        )                                                                                                                                                                                                                      =                              1                +                                                                            g                      ⁡                                              (                                                  1                          -                          α                                                )                                                              ⁢                                                                  z                                                  -                                                      (                                                          m                              +                              1                                                        )                                                                                              ⁡                                              (                                                  1                          -                                                                                    α                              k                                                        ⁢                                                          z                                                              -                                k                                                                                                                                    )                                                                                                                        (                                              1                        -                                                  α                          k                                                                    )                                        ⁢                                          {                                                                        (                                                      1                            -                                                          α                              ⁢                                                              xe2x80x83                                                            ⁢                                                              z                                                                  -                                  1                                                                                                                                              )                                                -                                                                                                            (                                                              1                                -                                α                                                            )                                                                                      (                                                              1                                -                                                                  α                                  k                                                                                            )                                                                                ⁢                                                                                    z                                                              -                                                                  (                                                                      m                                    +                                    1                                                                    )                                                                                                                      ⁡                                                          (                                                              1                                -                                                                                                      α                                    k                                                                    ⁢                                                                      z                                                                          -                                      k                                                                                                                                                                  )                                                                                                                          }                                                                                                                                              =                                                                    (                                          1                      -                                              α                        ⁢                                                  xe2x80x83                                                ⁢                                                  z                                                      -                            1                                                                                                                )                                    -                                                                                    (                                                  1                          -                          α                                                )                                                                    (                                                  1                          -                                                      α                            k                                                                          )                                                              ⁢                                                                  z                                                  -                                                      (                                                          m                              +                              1                                                        )                                                                                              ⁡                                              (                                                  1                          -                                                                                    α                              k                                                        ⁢                                                          z                                                              -                                k                                                                                                                                    )                                                                              +                                      g                    ⁢                                                                  (                                                  1                          -                          α                                                )                                                                    (                                                  1                          -                                                      α                            k                                                                          )                                                              ⁢                                                                  z                                                  -                                                      (                                                          m                              +                              1                                                        )                                                                                              ⁡                                              (                                                  1                          -                                                                                    α                              k                                                        ⁢                                                          z                                                              -                                k                                                                                                                                    )                                                                                                                                  (                                          1                      -                                              α                        ⁢                                                  xe2x80x83                                                ⁢                                                  z                                                      -                            1                                                                                                                )                                    -                                                                                    (                                                  1                          -                          α                                                )                                                                    (                                                  1                          -                                                      α                            k                                                                          )                                                              ⁢                                                                  z                                                  -                                                      (                                                          m                              +                              1                                                        )                                                                                              ⁡                                              (                                                  1                          -                                                                                    α                              k                                                        ⁢                                                          z                                                              -                                k                                                                                                                                    )                                                                                                                                                                    =                                                                    (                                          1                      -                                              α                        ⁢                                                  xe2x80x83                                                ⁢                                                  z                                                      -                            1                                                                                                                )                                    +                                                            (                                              g                        -                        1                                            )                                        ⁢                                                                  (                                                  1                          -                          α                                                )                                                                    (                                                  1                          -                                                      α                            k                                                                          )                                                              ⁢                                                                  z                                                  -                                                      (                                                          m                              +                              1                                                        )                                                                                              ⁡                                              (                                                  1                          -                                                                                    α                              k                                                        ⁢                                                          z                                                              -                                k                                                                                                                                    )                                                                                                                                  (                                          1                      -                                              α                        ⁢                                                  xe2x80x83                                                ⁢                                                  z                                                      -                            1                                                                                                                )                                    -                                                                                    (                                                  1                          -                          α                                                )                                                                    (                                                  1                          -                                                      α                            k                                                                          )                                                              ⁢                                                                  z                                                  -                                                      (                                                          m                              +                              1                                                        )                                                                                              ⁡                                              (                                                  1                          -                                                                                    α                              k                                                        ⁢                                                          z                                                              -                                k                                                                                                                                    )                                                                                                                                                    (        7        )            
Assuming a cyclic disturbance is
xcexd(z)=ejxcfx89t
and then z-transforming this equation gives                               V          ⁡                      (            z            )                          =                              Z            ⁡                          (                              v                ⁡                                  (                  t                  )                                            )                                =                      1                          1              -                                                z                                      -                    1                                                  ⁢                                  ⅇ                                      jω                    ⁢                                          xe2x80x83                                        ⁢                    T                                                                                                          (        8        )            
where xcfx89 is the angular frequency of the cyclic disturbance.
If we define a frequency transfer function "PHgr"g(z) as                                           Φ            g                    ⁡                      (            z            )                          =                  1                      1            +                                          HP                ⁡                                  (                  z                  )                                            ⁢                              G                ⁡                                  (                  z                  )                                                                                        (        9        )            
then
C(z)="PHgr"g(z)V(z)
is derived from equation (4) above.
From equations (7) and (9), we obtain                                                         Φ              g                        ⁡                          (              z              )                                =                                                    A                ⁡                                  (                  z                  )                                            -                              B                ⁡                                  (                  z                  )                                                                                    A                ⁡                                  (                  z                  )                                            +                                                (                                      g                    -                    1                                    )                                ⁢                                  B                  ⁡                                      (                    z                    )                                                                                      ⁢                  
                ⁢        where                            (        10        )                                          A          ⁡                      (            z            )                          =                  1          -                      α            ⁢                          xe2x80x83                        ⁢                          z                              -                1                                                                        (        11        )                                          B          ⁡                      (            z            )                          =                                            (                              1                -                α                            )                                      (                              1                -                                  α                  k                                            )                                ⁢                                    z                              -                                  (                                      m                    +                    1                                    )                                                      ⁡                          (                              1                -                                                      α                    k                                    ⁢                                      z                                          -                      k                                                                                  )                                                          (        12        )            
If we square the absolute value of equation (10), we obtain                                                                         |                                                      Φ                    g                                    ⁡                                      (                    z                    )                                                  ⁢                                  |                  2                                            =                                                                                          A                      ⁡                                              (                        z                        )                                                              -                                          B                      ⁡                                              (                        z                        )                                                                                                                        A                      ⁡                                              (                        z                        )                                                              +                                                                  (                                                  g                          -                          1                                                )                                            ⁢                                              B                        ⁡                                                  (                          z                          )                                                                                                                    ⁢                                  xe2x80x83                                ⁢                                                                                                    A                        ⁡                                                  (                          z                          )                                                                    _                                        -                                                                  B                        ⁡                                                  (                          z                          )                                                                    _                                                                                                                          A                        ⁡                                                  (                          z                          )                                                                    _                                        +                                                                  (                                                  g                          -                          1                                                )                                            ⁢                                                                        B                          ⁡                                                      (                            z                            )                                                                          _                                                                                                                                                                    =                                                |                                      A                    ⁡                                          (                      z                      )                                                        ⁢                                      |                    2                                    ⁢                                                            -                                              (                                                                                                            B                              ⁡                                                              (                                z                                )                                                                                      ⁢                                                                                          A                                ⁡                                                                  (                                  z                                  )                                                                                            _                                                                                +                                                                                    A                              ⁡                                                              (                                z                                )                                                                                      ⁢                                                                                          B                                ⁡                                                                  (                                  z                                  )                                                                                            _                                                                                                      )                                                              +                                    |                                      B                    ⁡                                          (                      z                      )                                                        ⁢                                      |                    2                                                                    |                                      A                    ⁡                                          (                      z                      )                                                        ⁢                                      |                    2                                    ⁢                                                                                    +                                                  (                                                      g                            -                            1                                                    )                                                                    ⁢                                              (                                                                                                            B                              ⁡                                                              (                                z                                )                                                                                      ⁢                                                                                          A                                ⁡                                                                  (                                  z                                  )                                                                                            _                                                                                +                                                                                    A                              ⁡                                                              (                                z                                )                                                                                      ⁢                                                                                          B                                ⁡                                                                  (                                  z                                  )                                                                                            _                                                                                                      )                                                              +                                                                  (                                                  g                          -                          1                                                )                                            2                                                        |                                      B                    ⁡                                          (                      z                      )                                                        ⁢                                      |                    2                                                                                                                          =                                                |                                      A                    ⁡                                          (                      z                      )                                                        ⁢                                      |                    2                                    ⁢                                                                                    -                        2                                            ⁢                                              {                                                                              R                            ⁢                                                          xe2x80x83                                                        ⁢                            e                            ⁢                                                          xe2x80x83                                                        ⁢                                                          (                                                              A                                (                                z                                )                                                            )                                                        ⁢                            R                            ⁢                                                          xe2x80x83                                                        ⁢                                                          e                              (                                                              B                                (                                z                                )                                                            )                                                                                +                                                      I                            ⁢                                                          xe2x80x83                                                        ⁢                                                          m                              (                                                              A                                (                                z                                )                                                            )                                                        ⁢                            I                            ⁢                                                          xe2x80x83                                                        ⁢                                                          m                              (                                                              B                                (                                z                                )                                                            )                                                                                                      }                                                              +                                    |                                      B                    ⁡                                          (                      z                      )                                                        ⁢                                      |                    2                                                                    |                                      A                    ⁡                                          (                      z                      )                                                        ⁢                                      |                    2                                    ⁢                                                                                    +                        2                                            ⁢                                              (                                                  g                          -                          1                                                )                                            ⁢                                              {                                                                              R                            ⁢                                                          xe2x80x83                                                        ⁢                            e                            ⁢                                                          xe2x80x83                                                        ⁢                                                          (                                                              A                                (                                z                                )                                                            )                                                        ⁢                            R                            ⁢                                                          xe2x80x83                                                        ⁢                                                          e                              (                                                              B                                (                                z                                )                                                            )                                                                                +                                                      I                            ⁢                                                          xe2x80x83                                                        ⁢                                                          m                              (                                                              A                                (                                z                                )                                                            )                                                        ⁢                            I                            ⁢                                                          xe2x80x83                                                        ⁢                                                          m                              (                                                              B                                (                                z                                )                                                            )                                                                                                      }                                                              +                                                                  (                                                  g                          -                          1                                                )                                            2                                                        |                                      B                    ⁡                                          (                      z                      )                                                        ⁢                                      |                    2                                                                                                          (        13        )            
Now let us assume that the cyclic disturbance defined by equation (8) is input to the system of FIG. 3. Then, the control output C(s) in a steady state is calculated as shown below from the final-value theorem.                                           lim                          t              →              ∞                                ⁢                      (                                          c                ⁡                                  (                  t                  )                                                            ⅇ                                  j                  ⁢                                      xe2x80x83                                    ⁢                  ω                  ⁢                                      xe2x80x83                                    ⁢                  t                                                      )                          =                                            lim                              z                →                1                                      ⁢                          (                                                (                                      1                    -                                          z                                              -                        1                                                                              )                                ⁢                                  Z                  ⁡                                      [                                                                  c                        ⁡                                                  (                          t                          )                                                                                            ⅇ                                                  j                          ⁢                                                      xe2x80x83                                                    ⁢                          ω                          ⁢                                                      xe2x80x83                                                    ⁢                          t                                                                                      ]                                                              )                                =                                    lim                              z                →                1                                      ⁢                          (                                                (                                      1                    -                                          z                                              -                        1                                                                              )                                ⁢                                  C                  ⁡                                      (                                                                  ⅇ                                                  j                          ⁢                                                      xe2x80x83                                                    ⁢                          ω                          ⁢                                                      xe2x80x83                                                    ⁢                          T                                                                    ⁢                      z                                        )                                                              )                                                              =                              lim                          z              →              1                                ⁢                      (                                          (                                  1                  -                                      z                                          -                      1                                                                      )                            ⁢                                                Φ                  g                                ⁡                                  (                                                            ⅇ                                              j                        ⁢                                                  xe2x80x83                                                ⁢                        ω                        ⁢                                                  xe2x80x83                                                ⁢                        t                                                              ⁢                    z                                    )                                            ⁢                              V                ⁡                                  (                                                            ⅇ                                              j                        ⁢                                                  xe2x80x83                                                ⁢                        ω                        ⁢                                                  xe2x80x83                                                ⁢                        t                                                              ⁢                    z                                    )                                                      )                                                  =                              lim                          z              →              1                                ⁢                      (                                          (                                  1                  -                                      z                                          -                      1                                                                      )                            ⁢                                                1                                      1                    -                                                                                            ⅇ                                                      j                            ⁢                                                          xe2x80x83                                                        ⁢                            ω                            ⁢                                                          xe2x80x83                                                        ⁢                            t                                                                          ⁡                                                  (                                                                                    ⅇ                                                              j                                ⁢                                                                  xe2x80x83                                                                ⁢                                ω                                ⁢                                                                  xe2x80x83                                                                ⁢                                t                                                                                      ⁢                            z                                                    )                                                                                            -                        1                                                                                            ·                                                      Φ                    g                                    ⁡                                      (                                                                  ⅇ                                                  j                          ⁢                                                      xe2x80x83                                                    ⁢                          ω                          ⁢                                                      xe2x80x83                                                    ⁢                          t                                                                    ⁢                      z                                        )                                                                        )                                                  =                                            lim                              z                →                1                                      ⁢                                          Φ                g                            ⁡                              (                                                      ⅇ                                          j                      ⁢                                              xe2x80x83                                            ⁢                      ω                      ⁢                                              xe2x80x83                                            ⁢                      t                                                        ⁢                  z                                )                                              =                                    Φ              g                        ⁡                          (                              ⅇ                                  j                  ⁢                                      xe2x80x83                                    ⁢                  ω                  ⁢                                      xe2x80x83                                    ⁢                  t                                            )                                          
This is because                               Z          ⁡                      (                                          ⅇ                                                      -                    α                                    ⁢                                      xe2x80x83                                    ⁢                  t                                            ⁢                              f                ⁡                                  (                  t                  )                                                      )                          =                  xe2x80x83                ⁢                              f            ⁡                          (              0              )                                +                                    ⅇ                                                -                  α                                ⁢                                  xe2x80x83                                ⁢                t                                      ⁢                          f              ⁡                              (                T                )                                      ⁢                          z                              -                1                                              +                                    ⅇ                                                -                  2                                ⁢                α                ⁢                                  xe2x80x83                                ⁢                t                                      ⁢                          f              ⁡                              (                                  2                  ⁢                  T                                )                                      ⁢                          z                              -                2                                              +          …          +                                                  xe2x80x83                ⁢                                            ⅇ                                                -                  n                                ⁢                                  xe2x80x83                                ⁢                α                ⁢                                  xe2x80x83                                ⁢                t                                      ⁢                          f              ⁡                              (                                  n                  ⁢                                      xe2x80x83                                    ⁢                  T                                )                                      ⁢                          z                              -                n                                              +          …                                        =                  xe2x80x83                ⁢                              f            ⁡                          (              0              )                                +                                    f              ⁡                              (                T                )                                      ⁢                                          (                                                      ⅇ                                          -                      αt                                                        ⁢                  z                                )                                            -                1                                              +                                    f              ⁡                              (                                  2                  ⁢                  T                                )                                      ⁢                                          (                                                      ⅇ                                          -                      αt                                                        ⁢                  z                                )                                            -                2                                              +          …          +                                                  xe2x80x83                ⁢                                            ⅇ                                                -                  n                                ⁢                                  xe2x80x83                                ⁢                a                ⁢                                  xe2x80x83                                ⁢                t                                      ⁢                          f              ⁡                              (                                  n                  ⁢                                      xe2x80x83                                    ⁢                  T                                )                                      ⁢                                          (                                                      ⅇ                                          -                      αt                                                        ⁢                  z                                )                                            -                n                                              +          …                                        =                  xe2x80x83                ⁢                  F          ⁡                      (                                          ⅇ                αt                            ⁢              z                        )                              
holds true in general, and therefore
Z(exe2x88x92xcex1tf(t))=F(excex1txc2x7z), (F(z)=Z(f(t)))
also holds true.
Consequently, the amplitude of the control output C(s), i.e., the amplitude of a frequency response, is given by the absolute value of the frequency transfer function, as shown below.
|"PHgr"g(ejxcfx89T)|
To find the value of |"PHgr"g(ejxcfx89T)|, the terms Re(A(ejxcfx89T)), Im(A(ejxcfx89T)), Re(B(ejxcfx89T)) and Im(B(ejxcfx89T)) must be calculated from equations (11) and (12), and then the calculations substituted into equation (13).
Since
A(ejxcfx89T)=1xe2x88x92xcex1exe2x88x92jxcfx89T=1xe2x88x92xcex1{cos(xcfx89T)xe2x88x92j sin(xcfx89T)}
from equation (11), we obtain
Re(A(ejxcfx89T))=1xe2x88x92xcex1 cos(xcfx89T)xe2x80x83xe2x80x83(14)
Im(A(ejxcfx89T))=xcex1 sin(xcfx89T)xe2x80x83xe2x80x83(15)
Since                               B          ⁡                      (                          ⅇ                              j                ⁢                                  xe2x80x83                                ⁢                ω                ⁢                                  xe2x80x83                                ⁢                t                                      )                          =                  xe2x80x83                ⁢                                            (                              1                -                α                            )                                      (                              1                -                                  α                  k                                            )                                ⁢                                    ⅇ                                                -                                      j                    ⁡                                          (                                              m                        +                        1                                            )                                                                      ⁢                ω                ⁢                                  xe2x80x83                                ⁢                T                                      ⁡                          (                              1                -                                                      α                    k                                    ⁢                                      ⅇ                                                                  -                        j                                            ⁢                                              xe2x80x83                                            ⁢                      k                      ⁢                                              xe2x80x83                                            ⁢                      ω                      ⁢                                              xe2x80x83                                            ⁢                      T                                                                                  )                                                              =                  xe2x80x83                ⁢                                            (                              1                -                α                            )                                      (                              1                -                                  α                  k                                            )                                ⁢                      [                          {                                                cos                  ⁡                                      (                                                                  (                                                  m                          +                          1                                                )                                            ⁢                      ω                      ⁢                                              xe2x80x83                                            ⁢                      T                                        )                                                  -                                  j                  ⁢                                      xe2x80x83                                    ⁢                                      sin                    ⁡                                          (                                                                        (                                                      m                            +                            1                                                    )                                                ⁢                        ω                        ⁢                                                  xe2x80x83                                                ⁢                        T                                            )                                                                                  }                        [                          1              -                                                α                  k                                ⁢                                  {                                                            cos                      ⁡                                              (                                                  k                          ⁢                                                      xe2x80x83                                                    ⁢                          ω                          ⁢                                                      xe2x80x83                                                    ⁢                          T                                                )                                                              -                                                                                                                                    xe2x80x83                    ⁢                      j            ⁢                          xe2x80x83                        ⁢                          sin              ⁡                              (                                  k                  ⁢                                      xe2x80x83                                    ⁢                  ω                  ⁢                                      xe2x80x83                                    ⁢                  T                                )                                              }                ]                                =                  xe2x80x83                ⁢                                            (                              1                -                α                            )                                      (                              1                -                                  α                  k                                            )                                [                                    {                                                                    cos                    ⁡                                          (                                                                        (                                                      m                            +                            1                                                    )                                                ⁢                        ω                        ⁢                                                  xe2x80x83                                                ⁢                        T                                            )                                                        ⁢                                      (                                          1                      -                                                                        α                          k                                                ⁢                                                  cos                          ⁡                                                      (                                                          k                              ⁢                                                              xe2x80x83                                                            ⁢                              ω                              ⁢                                                              xe2x80x83                                                            ⁢                              T                                                        )                                                                                                                )                                                  +                                                      α                    k                                    ⁢                                      sin                    ⁡                                          (                                                                        (                                                      m                            +                            1                                                    )                                                ⁢                        ω                        ⁢                                                  xe2x80x83                                                ⁢                        T                                            )                                                        ⁢                                      sin                    ⁡                                          (                                              k                        ⁢                                                  xe2x80x83                                                ⁢                        ω                        ⁢                                                  xe2x80x83                                                ⁢                        T                                            )                                                                                  }                        +                                                            xe2x80x83                ⁢                  j          ⁢                      {                                                            α                  k                                ⁢                                  cos                  ⁡                                      (                                                                  (                                                  m                          +                          1                                                )                                            ⁢                      ω                      ⁢                                              xe2x80x83                                            ⁢                      T                                        )                                                  ⁢                                  sin                  ⁡                                      (                                          k                      ⁢                                              xe2x80x83                                            ⁢                      ω                      ⁢                                              xe2x80x83                                            ⁢                      T                                        )                                                              -                                                sin                  ⁡                                      (                                                                  (                                                  m                          +                          1                                                )                                            ⁢                      ω                      ⁢                                              xe2x80x83                                            ⁢                      T                                        )                                                  ⁢                                  (                                      1                    -                                                                  α                        k                                            ⁢                                              cos                        ⁡                                                  (                                                      k                            ⁢                                                          xe2x80x83                                                        ⁢                            ω                            ⁢                                                          xe2x80x83                                                        ⁢                            T                                                    )                                                                                                      )                                                      }                          ]                                =                  xe2x80x83                ⁢                                            (                              1                -                α                            )                                      (                              1                -                                  α                  k                                            )                                [                                    {                                                cos                  ⁡                                      (                                                                  (                                                  m                          +                          1                                                )                                            ⁢                      ω                      ⁢                                              xe2x80x83                                            ⁢                      T                                        )                                                  -                                                      a                    k                                    ⁢                                      cos                    ⁡                                          (                                                                        (                                                      k                            +                            m                            +                            1                                                    )                                                ⁢                        ω                        ⁢                                                  xe2x80x83                                                ⁢                        T                                            )                                                                                  }                        +                                                            xe2x80x83                ⁢                  j          ⁢                      {                                          -                                  sin                  ⁡                                      (                                                                  (                                                  m                          +                          1                                                )                                            ⁢                      ω                      ⁢                                              xe2x80x83                                            ⁢                      T                                        )                                                              +                                                a                  k                                ⁢                                  sin                  ⁡                                      (                                                                  (                                                  k                          +                          m                          +                          1                                                )                                            ⁢                      ω                      ⁢                                              xe2x80x83                                            ⁢                      T                                        )                                                                        }                          ]            
also holds true from equation (12), we obtain                                                                         Re                ⁡                                  (                                      B                    ⁡                                          (                                              ⅇ                                                  i                          ⁢                                                      xe2x80x83                                                    ⁢                          ω                          ⁢                                                      xe2x80x83                                                    ⁢                          T                                                                    )                                                        )                                            =                              xe2x80x83                            ⁢                                                                    (                                          1                      -                      α                                        )                                                        (                                          1                      -                                              α                        k                                                              )                                                  ⁢                                  {                                                            cos                      ⁡                                              (                                                                              (                                                          m                              +                              1                                                        )                                                    ⁢                          ω                          ⁢                                                      xe2x80x83                                                    ⁢                          T                                                )                                                              -                                                                                                                                          xe2x80x83                            ⁢                                                                    a                    k                                    ⁢                                      cos                    ⁡                                          (                                                                        (                                                      k                            +                            m                            +                            1                                                    )                                                ⁢                        ω                        ⁢                                                  xe2x80x83                                                ⁢                        T                                            )                                                                      }                                                                        (        16        )                                                                                    Im                ⁡                                  (                                      B                    ⁡                                          (                                              ⅇ                                                  i                          ⁢                                                      xe2x80x83                                                    ⁢                          ω                          ⁢                                                      xe2x80x83                                                    ⁢                          T                                                                    )                                                        )                                            =                              xe2x80x83                            ⁢                                                                    (                                          1                      -                      α                                        )                                                        (                                          1                      -                                              α                        k                                                              )                                                  ⁢                                  {                                                            -                                              sin                        ⁡                                                  (                                                                                    (                                                              m                                +                                1                                                            )                                                        ⁢                            ω                            ⁢                                                          xe2x80x83                                                        ⁢                            T                                                    )                                                                                      +                                                                                                                                          xe2x80x83                            ⁢                                                                    a                    k                                    ⁢                                      sin                    ⁡                                          (                                                                        (                                                      k                            +                            m                            +                            1                                                    )                                                ⁢                        ω                        ⁢                                                  xe2x80x83                                                ⁢                        T                                            )                                                                      }                                                                        (        17        )            
By substituting equations (14) to (17) into equation (13), we can find the value of
|"PHgr"g(ejxcfx89T)|
which is the absolute value of the frequency transfer function, i.e., the amplitude of a frequency response.
FIG. 4 is the result of calculating the amplitude of frequency response |"PHgr"g(ejxcfx89T)| in a case when the control gain ratio g is used as the parameter and the value thereof is varied from 1.0 to 0.2 in units of 0.2, thereby increasing the control gain K* on a step-by-step basis. In the figure, the horizontal axis represents the period of cyclic disturbance and the vertical axis represents the amplitude of frequency response |"PHgr"g(ejxcfx89T)|. In this case, we assume that the dead time=300 sec, time constant=480 sec, sampling time=30 sec, control gain K*=1/g, k=10, and settling time=(k+m)T=600 sec.
In FIG. 4, periods of disturbance in which the amplitude of frequency response |"PHgr"g(ejxcfx89T)| is smaller than 1 mean that the amplitude decreases as the result of feedback control. Conversely, periods of disturbance in which the amplitude of frequency response |"PHgr"g(ejxcfx89T)| is greater than 1 mean that the amplitude increases as the result of feedback control.
For example, if g=1.0, the amplitude of cyclic disturbance with a period of 15 minutes increases by a factor of 1.8. If g=0.4, i.e., the control gain is made 2.5 times larger, then the amplitude increases only by a factor of 1.2 at the most. Since the amplitude |"PHgr"g(ejxcfx89T)| of cyclic disturbance with that period is greater than 1.0, system performance will never be improved no matter how much the control gain is increased. This is a limitation of feedback control.
On the other hand, if g=1.0, cyclic disturbance with a period of 100 minutes is attenuated down to half. Feedback control thus exerts its effect. If g=0.4, however, the disturbance is attenuated only by a factor of 0.9. Thus, system performance improvements by feedback control can hardly be expected. Accordingly, it is now understood that the effect of feedback control greatly varies depending on the combination of the period of presence of a disturbance and a control gain.
The present invention provides a method for controlling a process by calculating a manipulated variable from a deviation between setpoint and control output variables and then using the manipulated variable, wherein the time-series data of the control output variable is subjected to a frequency analysis, the rate of change in the frequency response amplitude of the control output variable in relation to a control gain ratio is determined, and a control gain is adjusted according to the rate of change and the results of the frequency analysis. As a result, it is possible to automatically set a control gain according to the condition of disturbance. Thus, optimum control can always be achieved and follow-up capability can be improved for such events as a setpoint change.